The above effective Hamiltonian works at the small x regime. We can see that the dynamics of doped excitons on the background of \(120^\circ \)  magnetic order is basically captured by a spin 1/2 hard core boson gas problem, an observation already made in our previous paper[1]}. Note that Eq. REF  can be viewed as a bosonic t-J model. Next we analyze this effective model. We will find a spin polarized BEC at small \(J/t\)  and a spin gapped paired superfluid at large \(J/t\) . In the intermediate regime there is a spiral phase where the boson \(b\)  condenses at non-zero momentum.
